Sometimes, but not always, you can write implicitely defined functions in terms of explicit functions of a parameter. For example, you can graph x2+ y2= 1, a circle, in parametric graphing mode with x= cos(t), y= sin(t).

Now the TI-89 is programmable so you might search the TI site for packages you can upload to expand its capacities or write your own. At worst you can graph the inequality f(x,y) < 0 by iterating over the coordinates on the visible screen and plotting the ones satisfying the inequality. The boundary will then be your curve.